Dimensional Transformation in Optical Communication

ABSTRACT

A transmitter ( 102,200 ) applies a dimensional transformation to preliminary digital drive signals representing symbols, thereby generating transformed digital drive signals ( 704 ) designed to represent each symbol using a plurality of first dimensions of an optical carrier ( 242 ), the first dimensions distributed over two or more timeslots. The preliminary digital drive signals are designed to represent each symbol using a plurality of second dimensions of the carrier, which differ from the first dimensions. Using the transformed signals, the transmitter generates ( 706 ) an optical signal ( 260 ). A receiver ( 102,300 ) receives ( 802 ) an optical signal ( 360 ) and determines received digital signals ( 804 ) corresponding to the first dimensions. The receiver applies an inverse dimensional transformation to the received digital signals to generate preliminary digital drive signal estimates ( 806 ) corresponding to the second dimensions, thereby permitting estimation of the symbols ( 808 ). The inverse dimensional transformation may average signal degradations in the received digital signals.

TECHNICAL FIELD

This document relates to the technical field of optical communications.

BACKGROUND

In an optical communications system, a transmitter may encode clientdata bits by mapping them to multi-bit symbols, and then using aparticular modulation scheme to modulate one or more optical carrierswith the symbols, thereby generating an optical signal to be transmittedvia a communications channel to a receiver, where the optical signal isrepresentative of digital information. The receiver may process anoptical signal received via the communications channel to recoverestimates of the multi-bit symbols, the client data bits, or both.

The optical signal received at the receiver may comprise a degradedversion of the optical signal that was generated at the transmitter.Various components of the optical communications system may contributeto signal degradation, including optical fibers, optical amplifiers,filters, isolators, and the like. Effects such as amplifier noise,optical nonlinearity, polarization dependent loss or gain (PDL), andpolarization mode dispersion (PMD) may introduce noise and/or distortioninto the signal. The amplitude of the noise relative to the amplitude ofthe optical signal may be characterized by the signal-to-noise ratio(SNR), or alternatively by the noise-to-signal ratio (NSR). The NSR maybe convenient when dissecting noise sources. A high NSR may result innoisy symbol estimates, which may in turn produce erroneous estimates ofthe client data bits. The probability that client data bit estimatesrecovered at the receiver will differ from the original client data bitsencoded at the transmitter may be characterized by the Bit Error Ratioor Bit Error Rate (BER). A given application may have a maximum BERtolerance. For example, an application may require that the BER notexceed 10⁻¹⁶.

Forward Error Correction (FEC) techniques may be used to reduce the BER.Instead of the transmitter mapping the original client data bitsdirectly to multi-bit symbols, the client data bits may first undergoFEC encoding based on a chosen FEC scheme. The resulting FEC-encodedbits include redundant information, such as parity or check bits. Thebit estimates recovered at the receiver will be estimates of theFEC-encoded bits that were generated the transmitter. These estimatesmay undergo FEC decoding at the receiver based on the chosen FEC scheme.The FEC decoding makes use of the redundant information that wasincluded in the FEC-encoded bits in order to detect and correct biterrors.

FEC encoding is advantageous in that it may permit error control withoutthe need to resend data packets. However, this is at the cost ofincreased overhead. The amount of overhead or redundancy added by FECencoding may be characterized by the information rate R, where R isdefined as the ratio of the amount of input information to the amount ofdata that is output after FEC encoding (which includes the overhead).For example, if FEC encoding adds 25% overhead, then for every fourinformation bits that are to be FEC-encoded, the FEC encoding will add 1bit of overhead, resulting in 5 FEC-encoded data bits to be transmittedto the receiver. This corresponds to an information rate R=4/5=0.8.

SUMMARY

According to a broad aspect, an optical receiver is operative to receivean optical signal over an optical communications channel establishedbetween the optical receiver and an optical transmitter, wherein thereceived optical signal comprises a degraded version of a modulatedoptical signal generated at the optical transmitter. The opticalreceiver is operative to determine received digital signalscorresponding to a plurality of first dimensions of the received opticalsignal, wherein the first dimensions correspond to dimensions of anoptical carrier modulated at the optical transmitter to represent amulti-bit symbol, and wherein the first dimensions are distributed overtwo or more timeslots. The optical receiver is operative to determinepreliminary digital drive signal estimates using an inverse dimensionaltransformation and the received digital signals, the preliminary digitaldrive signal estimates corresponding to a plurality of seconddimensions. The optical receiver is operative to determine an estimateof the multi-bit symbol using the preliminary digital drive signalestimates.

According to some examples, the plurality of second dimensions is lessthan the plurality of first dimensions.

According to some examples, the two or more timeslots may be consecutiveor non-consecutive.

According to some examples, the plurality of first dimensions isdistributed over two polarizations.

According to some examples, the plurality of first dimensions isdistributed over in-phase (I) and quadrature (Q) components of at leastone polarization.

According to some examples, the inverse dimensional transformationaverages signal degradations in the received digital signals, the signaldegradations caused by one or more of noise, nonlinear effects,polarization dependent loss or gain (PDL), and analog imperfections.

According to some examples, the inverse dimensional transformationcomprises a matrix, wherein the matrix is substantially linear andsubstantially unitary.

According to some examples, the received optical signal is processedusing an adaptive equalization circuit to compensate for linearimpairments in the optical communications channel.

According to a broad aspect, an optical transmitter is operative togenerate preliminary digital drive signals representative of multi-bitsymbols. The optical transmitter is operative to generate transformeddigital drive signals from the preliminary digital drive signals,wherein the transformed digital drive signals are designed to representeach multi-bit symbol using a plurality of first dimensions of anoptical carrier, the first dimensions being distributed over two or moredistinct timeslots. The preliminary digital drive signals are designedto represent each multi-bit symbol using a plurality of seconddimensions of the optical carrier, the second dimensions differing fromthe first dimensions. The optical transmitter is operative to use thetransformed digital drive signals to generate an optical signal fortransmission over an optical communications channel established betweenthe optical transmitter and an optical receiver.

According to some examples, the plurality of second dimensions is lessthan the plurality of first dimensions.

According to some examples, the two or more timeslots may be consecutiveor non-consecutive.

According to some examples, the plurality of first dimensions isdistributed over two polarizations.

According to some examples, the plurality of first dimensions isdistributed over in-phase (I) and quadrature (Q) components of at leastone polarization.

According to some examples, the transformed digital drive signals aregenerated by applying a dimensional transformation to the preliminarydigital drive signals. The dimensional transformation may comprise amatrix. The matrix may be substantially linear and substantiallyunitary.

According to some examples, the transformed digital drive signals aregenerated from the preliminary digital drive signals using alook-up-table.

According to some examples, the optical transmitter is operative toapply frequency-domain processing to the transformed digital drivesignals. The frequency-domain processing may comprise applying a matchedfilter to the transformed digital drive signals.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates an example optical communications system inaccordance with the technology disclosed herein;

FIG. 2 illustrates an example transmitter in accordance with thetechnology disclosed herein;

FIG. 3 illustrates an example receiver in accordance with the technologydisclosed herein;

FIG. 4 illustrates a plot of bit error rate (BER) as a function of thelinear noise-to-signal ratio (NSR) for a 64-level quadrature amplitudemodulation (64-QAM) scheme;

FIG. 5 illustrates a magnified portion of the plot illustrated FIG. 4with example points A and B;

FIG. 6 illustrates the second derivative of the BER in FIG. 4 withrespect to the NSR, plotted as a function of BER;

FIG. 7 illustrates an example method for implementing a dimensionaltransformation at a transmitter;

FIG. 8 illustrates an example method for implementing an inversedimensional transformation at a receiver;

FIG. 9 is a schematic diagram illustrating the implementation of adimensional transformation at a transmitter according to a firstexample;

FIG. 10 is a schematic diagram illustrating example details forimplementing the dimensional transformation according to the firstexample;

FIG. 11 is a schematic diagram illustrating the implementation of aninverse dimensional transformation at a receiver according to the firstexample;

FIG. 12 is a schematic diagram illustrating example details forimplementing the inverse dimensional according to the first example;

FIG. 13 is a schematic diagram illustrating the implementation of adimensional transformation at a transmitter according to a secondexample;

FIG. 14 is a schematic diagram illustrating the implementation of aninverse dimensional transformation at a receiver according to the secondexample; and

FIG. 15 is a histogram of received values which have undergone aninverse dimensional transformation according to a fifth example.

DETAILED DESCRIPTION

FIG. 1 illustrates an optical communication system 100 in accordancewith the technology disclosed herein. The communications system 100comprises transceivers 102. An optical signal, representative of digitalinformation (also referred to as client data), is transmitted betweenthe transceivers 102 via an optical communications channel 104. Thetransceivers 102 may be flexible, such that various configurationparameters of the transceivers 102 can be adjusted. For the opticalcommunication system 100 to be operable, the configuration parameters ofa transmitter section of one of the transceivers 102 must be compatiblewith the configuration parameters of a receiver section of the other ofthe transceivers 102. Examples of configuration parameters include amodulation format or scheme, symbol rate, forward error correction (FEC)parameters, digital signal processing (DSP) parameters, pulse shapingparameters, the number of sub-carriers for frequency divisionmultiplexing (FDM), chromatic dispersion compensation parameters,carrier phase recovery parameters, and digital nonlinear compensationparameters.

For the purposes of the present disclosure, it is convenient to considera transmitted optical signal, such as the signal transmitted via theoptical communications channel 104, as a function of four orthogonaldimensions versus time. The four orthogonal dimensions comprise therespective in-phase (I) and quadrature (Q) components of each of twoorthogonal polarizations, denoted generically as X and Y. Forsimplicity, the polarizations at the transmitter, which are linear, maybe denoted as X_(Tx) and Y_(Tx), respectively. These orthogonalpolarizations rotate along the optical path from the transmitter to thereceiver, and are generally elliptical in shape. For notation purposes,the four dimensions at a particular timeslot, t, may be denoted asXI(t), XQ(t), YI(t), and YQ(t). At a different timeslot, t+T, the fourdimensions of the optical signal may be denoted as XI(t+T), XQ(t+T),YI(t+T), and YQ(t+T). When the dimensions of the optical signal at thetwo different timeslots, t and t+T, are combined, the total number ofdimensions resulting from the combination would be eight, and thesedimensions would be denoted as: XI(t), XQ(t), YI(t), YQ(t), XI(t+T),XQ(t+T), YI(t+T), and YQ(t+T).

A signal transmitted via the optical communications channel 104 may bealtered by various elements of the optical communications system, suchas optical fibers, optical amplifiers, filters, isolators,wavelength-selective switches, and the like. For example, the passage ofa signal through an optical fiber or an optical filter may attenuate theoptical signal, whereas the passage of the signal through an opticalamplifier may strengthen the signal. The signal loss (or signal gain)caused by a given component may depend on the polarization state of thesignal. In general, this effect is referred to as polarization dependentloss or gain (denoted PDL). Where two channels of information aretransmitted on the same carrier frequency using waves of two orthogonalpolarization states, denoted as X_(PDL) and Y_(PDL), a given element ofthe system may cause each channel to experience a different level ofPDL. PDL is cumulative across all elements in the optical communicationssystem. As a result of PDL, one polarization may be noisier than anotherpolarization.

Random imperfections in an optical fiber may cause two orthogonalpolarizations to propagate with different speeds. This effect, referredto as polarization mode dispersion (PMD), causes the two polarizationcomponents of a signal, denoted as X_(PMD) and Y_(PMD), to slowlyseparate over the length of an optical fiber, thereby causing pulses tobroaden and overlap. The PMD of a signal may be characterized by anumber, M, of timeslots over which the overlapping occurs. M may also bereferred to as the PMD “memory”. PMD compensation may be achieved usingan adaptive filter at the receiver, such as a least means squares (LMS)circuit. However, an LMS circuit may add a correlation between noisecomponents of the symbols at different times. This noise correlation maybe observed in the same M timeslots in which PMD memory is observed.

Measurement and mitigation techniques for PDL and/or PMD are described,for example, in U.S. Pat. No. 7,305,183 to Roberts et al.; U.S. Pat. No.7,382,985 to Roberts et al.; U.S. Pat. No. 7,532,822 to Sun et al.; U.S.Pat. No. 7,936,999 to Hawryluck et al.; U.S. Pat. No. 8,385,747 toRoberts et al.; U.S. Pat. No. 8,718,491 to Khandani et al.; U.S. Pat.No. 9,602,207 to Khandani et al.; and in the following publications:Mumtaz et al. “PDL Mitigation in PolMux OFDM Systems Using Golden andSilver Polarization-Time Codes,” Optical Fiber Communication Conference,OSA Technical Digest (CD) (Optical Society of America, 2010), paperJThA7; Mumtaz et al. “Space-Time codes for optical fiber communicationwith polarization multiplexing,” IEEE International Conference onCommunications (IEEE, 2010), pp. 1-5; and Meron et al. “Use ofspace-time coding in coherent polarization-multiplexed systems sufferingfrom polarization-dependent loss,” Opt. Lett. 35(21), 3547-3549 (2010),each of which is incorporated by reference herein.

U.S. Pat. Nos. 8,718,491 and 9,602,207 describe the application of anoise whitening matrix to both reduce the total noise and to make thenoise variances equal between orthogonal polarizations. The noisewhitening matrix is only applied at the receiver, and may be dynamicallyupdated in response to any changes in the optical line. A transmit Jonesrotation matrix may be applied at the transmitter, in which the rotationangles attempt to track the changes the optical line such that thereceived orientation is optimum relative to the PDL of the noise.

The publications by Mumtaz et al. and Meron et al. describe gold andsilver space-time codes which may be used to mitigate the effects ofPDL. The implementation of gold and silver codes generally requiresintricate decoding circuits.

FIG. 2 is a block diagram illustration of an example transmitter section200 of a transceiver (“transmitter 200”), in accordance with examples ofthe technology disclosed herein.

The transmitter 200 is operative to transmit an optical signal 260 whichis representative of client data bits 204. The transmitter 200 employspolarization-division multiplexing (PDM). In other examples (not shown),generation of the optical signal may involve alternative techniques,such as single polarization modulation, modulation of an unpolarizedcarrier, mode-division multiplexing, spatial-division multiplexing,Stokes-space modulation, polarization balanced modulation, and the like.A laser 240 is operative to generate a continuous wave (CW) opticalcarrier 242. A polarizing beam splitter 244 is operative to split theoptical carrier 242 into orthogonally-polarized components 246, 248 thatare modulated by respective electrical-to-optical modulators 250, 252 toproduce modulated polarized optical signals 254, 256 that are combinedby a beam combiner 258, thus yielding an optical signal 260.

An application-specific integrated circuit (ASIC) 202 is operative toproduce I and Q analog drive signals 232, 234 to drive theelectrical-to-optical modulator 250. The ASIC 202 is operative toproduce I and Q analog drive signals 236, 238 to drive theelectrical-to-optical modulator 252.

The ASIC 202 may be operative to apply FEC encoding 206 to the clientdata bits 204, thereby generating FEC-encoded bits 208. The FEC-encodedbits 208 may be mapped to multi-bit symbols in accordance with aspecific code, as denoted by bit-to-symbol mapping 210. Thebit-to-symbol mapping 210 may produce a stream of multi-bit symbols 212.

The ASIC 202 further comprises a transmit digital signal processor (DSP)214 and a plurality of digital-to-analog converters (DACs). The transmitDSP 214 is operative to process the symbols 212, for example, byperforming one or more of pulse shaping, subcarrier multiplexing,chromatic dispersion precompensation, and distortion precompensation onthe symbols. The processing performed by the transmit DSP 214 mayinclude the application of one or more filters, which may involve theapplication of one or more Fast Fourier Transforms (FFTs) and one ormore corresponding inverse FFTs (IFFTs).

Based on the symbols 212 and a selected modulation scheme, the transmitDSP 214 is operative to generate four digital drive signals at aparticular timeslot, t, corresponding to the four dimensions XI, XQ, YI,YQ. For example, digital drive signals 216, 218 may correspond to the Iand Q components, respectively, of the X polarization, while digitaldrive signals 220, 222 may correspond to the I and Q components,respectively, of the Y polarization. According to this example, at thetimeslot, t, the digital drive signals 216, 218 may be denotedS_(XI)(t), S_(XQ)(t), respectively, while digital drive signals 220, 222may be denoted S_(YI)(t), S_(YQ)(t), respectively.

The digital drive signals 216, 218, 220, 222 may be converted byrespective DACs 224, 226, 228, 230 into the analog drive signals 232,234, 236, 238, respectively. As previously described, the analog drivesignals 232, 234, 236, 238 are used to drive the electrical-to-opticalmodulators 250, 252, which ultimately results in the optical signal 260.

The transmitter 200 may comprise additional components that are notdescribed in this document.

FIG. 3 is a block diagram illustration of an example receiver section ofa transceiver (“receiver 300”), in accordance with examples of thetechnology disclosed herein.

The receiver 300 is operative to recover corrected client data bits 304from a received optical signal 360. The received optical signal 360 maycomprise a degraded version of the optical signal 260 generated by thetransmitter 200, where the degradations in the received optical signal360 may have been caused, for example, by one or more of noise,nonlinear effects, PDL, and imperfections in analog signal processingperformed at the transmitter 200. A polarizing beam splitter 344 isoperative to split the received optical signal 360 intoorthogonally-polarized components 354, 356. An optical hybrid 358 isoperative to process the components 354, 356 with respect to an opticalsignal 342 produced by a laser 340. Photodetectors 362 are operative toconvert the outputs 346, 348, 350, 352 of the optical hybrid 358 toreceived analog signals 332, 334, 336, 338, respectively. The fourreceived analog signals correspond to the four dimensions XI, XQ, YI, YQat a particular timeslot, t.

An ASIC 302 comprises analog-to-digital converters (ADCs) 324, 326, 328,330 which are operative to sample the received analog signals 332, 334,336, 338, respectively, and to generate received digital signals 316,318, 320, 322, respectively. In one example, the received analog signals332, 334 may correspond to the I and Q components, respectively, of theX polarization, while the received analog signals 336, 338 maycorrespond to the I and Q components, respectively, of the Ypolarization. According to this example, at the timeslot, t, thereceived digital signals 316, 318 may be denoted R_(XI)(t), R_(XQ)(t),respectively, while the received digital signals 320, 322 may be denotedR_(YI)(t) and R_(YQ)(t), respectively.

The ASIC 302 comprises a receive DSP 314 which is operative to processthe received digital signals 316, 318, 320, 322. For example, thereceive DSP 214 may be operative to apply one or more filters to thedigital signals 316, 318, 320, 322, which may involve the application ofone or more FFTs and one or more corresponding IFFTs. The receive DSP314 may output digital signals 370, 372, 374, 376 based on the digitalsignals 316, 318, 320, 322.

The ASIC 302 is operative to apply a carrier recovery process 313 to thedigital signals 370, 372, 374, 376 in order to derive symbol estimates312 for the two orthogonal polarizations. The symbol estimates 312 areestimates of the symbols 212 that were generated by the bit-to-symbolmapping 210 performed at the transmitter 200.

The ASIC 302 is operative to apply symbol-to-bit demapping 310 to thesymbol estimates 312 in order to derive bit estimates 308. Thesymbol-to-bit demapping 310 involves applying an inverse of the codethat was used in the bit-to-symbol mapping 210. The bit estimates 308are estimates of the bits 208 that were generated by the FEC encoding206 at the transmitter 200. A bit estimate may comprise a binary value,or may comprise a confidence value, such as log-likelihood ratio. In thecase of a binary value (i.e., a bit), log-likelihood ratio (LLR) isdefined as the logarithm of the ratio of the probability of the bitbeing equal to one to the probability of the bit being equal to zero.For example, for a bit “b”, LLR(b)=log P(b=1)/P(b=0), where P denotesprobability. For non-binary values, such as a set of integers, othermetrics could be used, such as the logarithm of the probability of agiven integer value divided by the sum of the probabilities of the otherpossible integer values, for example.

The ASIC 302 is operative to apply FEC decoding 306 to the bit estimates308 in order to recover the corrected client data bits 304. The FECdecoding 306 may comprise hard-decision decoding or soft-decisiondecoding. One example of soft-decision decoding is Maximum Likelihood(ML) decoding. If the FEC decoding 306 is able to correct all errorspresent in the FEC-encoded bit estimates 308, then the corrected clientdata bits 304 will be identical to the original client data bits 204. Ifthe FEC decoding 306 is unable to correct all errors present in theFEC-encoded bit estimates 308, then the corrected client data bits 304will differ from the original client data bits 204. In this case, theFEC scheme chosen for the FEC encoding 206 and FEC decoding 306 will beconsidered to have failed.

The receiver 300 may comprise additional components that are notdescribed in this document.

The success or failure of a given FEC scheme depends on its strengthrelative to the extent of the errors present in the FEC-encoded bitestimates. FEC decoding will generally respond to the average BER of theFEC-encoded bit estimates to which it is applied. The average BERobserved at the input of the FEC decoding may be denoted BER_(FEC_AVG).Hard decision FEC decoding may be unable to correct all of the errors inthe FEC-encoded bit estimates when BER_(FEC_AVG) exceeds some threshold,denoted BER_(THRESH). In other words, the FEC scheme used for the FECencoding at the transmitter and the FEC decoding at the receiver isexpected to fail when BER_(FEC_AVG)>BER_(THRESH). In one example,BER_(THRESH) is on the order of 3.84×10⁻¹.

Generally, the BER of the FEC-encoded bit estimates 308 is expected toincrease as the noise in the received optical signal 360 increases. Theprecise relationship between the noise-to-signal ratio (NSR) and BERdepends on the code used for the bit-to-symbol mapping 210 andmodulation scheme used by the transmit DSP 214 to convert the symbols212 to the digital drive signals 216, 218, 220, 222, and also on theshape of the four-dimensional probability density function of the noise,in the event that the noise is not isotropic Gaussian noise.

FIG. 4 illustrates a plot of BER as a function of the linear NSR for a64-level quadrature amplitude modulation (64-QAM) scheme.

There may be circumstances in which different streams of bits (orsymbols) experience different noise levels. For example, as describedpreviously, PDL may cause different polarizations to have differentnoise levels. Thus, for example, symbols transmitted in the Xpolarization may exhibit a different level of the noise than symbolstransmitted in the Y polarization. It follows that the FEC-encoded bitestimates determined from one symbol stream may have a different BERthan the FEC-encoded bit estimates determined from another symbolstream.

One may consider a simple example in which a first set of FEC-encodedbit estimates exhibits a first BER, denoted BER_(A), and a second set ofFEC-encoded bit estimates exhibits a second BER, denoted BER_(B), whereBER_(A)≠BER_(B). If the number of FEC-encoded bit estimates in each setis equal, the average BER across the two sets would beBER_(FEC_AVG)=(BER_(A)+BER_(B))/2. If hard decision FEC decoding isapplied to the two sets, the FEC scheme is expected to fail ifBER_(FEC_AVG)=(BER_(A)+BER_(B))/2 exceeds BER_(THRESH) for the FECscheme. This is because the performance of the FEC depends on theaverage BER of the FEC-encoded bit estimates to which it is applied.

The differing BER values of the FEC-encoded bit estimates are the resultof differing noise levels in the symbol estimates from which theFEC-encoded bit estimates were determined. As an alternative to applyingFEC decoding to sets of bits that exhibit the different BERs, there maybe advantages to achieving a more uniform noise level across the symbolestimates, such that the FEC-encoded bit estimates determined from thesymbol estimates have a more uniform BER. A more uniform noise level maybe achieved across all symbol estimates by averaging the different noiselevels exhibited by different groups of symbol estimates. Examples ofhow this noise averaging may be achieved will be described in detailwith respect to FIGS. 7-15.

Where a noise averaging technique has been applied, the symbol estimatesgenerated at the receiver may have a substantially uniform noise level,such that the resulting FEC-encoded bit estimates have a substantiallyuniform BER, which may be denoted BER_(NOISE_AVG). In contrast toBER_(FEC_AVG), which is determined by directly averaging BER_(A) andBER_(B). BER_(NOISE_AVG) is determined using the relationship betweenBER and symbol noise for the specific modulation scheme being used. Forexample, FIG. 5 illustrates a magnified portion of the plot illustratedin FIG. 4, with example points A and B denoting two sets of symbolestimates having two different noise levels which are associated withBER_(A) and BER_(B), respectively. As illustrated in FIG. 5, acalculation of BER_(FEC_AVG) may be represented schematically by drawinga straight line between points A and B on the curve, and thendetermining the BER that corresponds to the center point of that line.In contrast, BER_(NOISE_AVG) may be determined by first determining theaverage linear NSR associated with BER_(A) and BER_(B), and then usingthe curve to determine the BER that corresponds to this average linearNSR. It is apparent from the magnified plot of FIG. 5, thatBER_(NOISE_AVG) is less than BER_(FEC_AVG). In other words, performingan operation that averages the noise across the two sets of symbols willresult in a uniform BER (BER_(NOISE_AVG)) that is less than the averageBER that the FEC scheme would be responding to (BER_(FEC_AVG)) if thenoise averaging operation is not performed.

It may be of interest to ensure that the bit estimates undergoing FECdecoding have a BER that is as low as possible, so as to reduce thelikelihood that the FEC decoding will fail, or to permit the use of ahigher rate FEC scheme that requires less overhead. Accordingly, for theexample points A and B in FIG. 5, it may be of interest to implement anoise averaging technique so that the FEC decoding only needs to respondto the lower value of BER_(NOISE_AVG), instead of the higher value ofBER_(FEC_AVG), that the FEC decoding would need to handle in the absenceof noise averaging.

However, there are other examples in which it may be of interest for theFEC decoding to handle bit estimates having a range of BERs, such thatthe FEC responds to BER_(FEC_AVG), rather than using noise averaging togenerate a uniform value of BER_(NOISE_AVG). Referring to FIG. 5,BER_(NOISE_AVG) is less than BER_(FEC_AVG) because the points A and Bare located in a convex region of the curve in FIG. 4. However, it maybe shown that there are other points on the curve, specifically thoselocated in the concave region of the curve in FIG. 4, for whichBER_(NOISE_AVG) is greater than BER_(FEC_AVG).

The convex and concave regions of the curve in FIG. 4 may be more easilydistinguished from one another by considering the second derivative ofBER with respect to linear NSR, which is plotted as a function of BER inFIG. 6. Those BER values for which the second derivative is positivecorrespond to the convex region of the curve in FIG. 4, whereas thoseBER values for which the second derivative is negative correspond to theconcave region of the curve in FIG. 4. As is apparent from FIG. 6, BERvalues of less than 0.025 are within the convex region, whereas BERvalues of greater than 0.025 are within the concave region. Although notexplicitly illustrated, it may be shown that, for two points located inthe concave region (i.e., corresponding to two different BER values,each greater than 0.025), applying a noise averaging operation mayresult in a single uniform BER value, BER_(NOISE_AVG), that is greaterthan the value of BER_(FEC_AVG) for the two points. This is one examplein which it may be preferable to let the FEC respond to BER_(FEC_AVG),rather than using noise averaging.

The choice of whether to perform noise averaging may depend on thedifferent noise levels (and BERs) in question. In U.S. patentapplication Ser. No. 15/672,434 filed on Aug. 9, 2017, Oveis-Gharan etal. describe a technique referred to as contrast coding, in which noiseis redistributed to generate different classes of bit estimates, whereeach class may be associated with a different average BER. Within agiven class, the effects of PDL may produce a range of BER values. Thechoice of whether to let the FEC decoding handle the range of BERvalues, or whether to instead perform a noise averaging operation maydepend on the average BER of the class. For example, a low-BER class mayinclude a range of BER values located within the convex portion of thecurve in FIG. 4. In this case, it may be advantageous to handle the PDLby using a noise averaging operation to obtain a substantially uniformBER value within the class. In another example, a high-BER class mayinclude a range of BER values located within the concave portion of thecurve in FIG. 4. In this case, it may be advantageous to handle the PDLby letting the FEC decoding respond directly to the range of BER valueswithin the class.

Returning to FIG. 2, the optical signal 260 is generated at thetransmitter 200 by modulating dimensions of the CW optical carrier 242to represent the stream of multi-bit symbols 212. The modulating isachieved using the digital drive signals 216, 218, 220, 222. In a simpleexample, a single multi-bit symbol may be represented in four dimensionsXI, XQ, YI, YQ at a single timeslot, t, by using the digital drivesignals S_(XI)(t), S_(XQ)(t), S_(YI)(t), S_(YQ)(t), respectively.

However, rather than restricting the dimensions used to represent themulti-bit symbol to a single timeslot, it may be advantageous for thosedimensions to be distributed over two or more distinct timeslots. Thesetimeslots may be consecutive or non-consecutive. The timeslots may bespread out over a longer time span based on interleaving. Byrepresenting each multi-bit symbol using dimensions that span aplurality of timeslots, it may be possible to average signaldegradations, including degradations caused by one or more of noise,nonlinear effects, PDL, and analog imperfections.

For the purposes of the following examples, the term “dimensionaltransformation” may be understood as an operation that results intransformed digital drive signals that are used at a transmitter tomodulate dimensions of an optical carrier to represent multi-bitsymbols. According to some examples, the transformed digital drivesignals resulting from the dimensional transformation modulate theoptical carrier such that each multi-bit symbol is represented using aplurality of first dimensions of the optical carrier, where the firstdimensions are distributed over two or more distinct timeslots.According to some examples, the transformed digital drive signals aregenerated as a result of applying the dimensional transformation topreliminary digital drive signals, the preliminary digital drive signalshaving been designed to modulate dimensions of the optical carrier torepresent multi-bit symbols according to a specific modulation scheme.According to some examples, the preliminary digital drive signals mayhave been designed to modulate the optical carrier such that eachmulti-bit symbol is represented using a plurality of second dimensions,where the plurality of second dimensions is less than the plurality offirst dimensions. In other words, the effect of the dimensionaltransformation may be to increase the number of dimensions over whicheach multi-bit symbol is represented, thereby resulting in transformeddigital drive signals that cause each multi-bit symbol to be representedby more dimensions than would be the case if the preliminary digitaldrive signals were used to modulate the optical carrier to representeach multi-bit symbol.

The dimensional transformation may be implemented as one or more serialsteps, as one or more parallel steps, or as a combination of both serialand parallel steps. In some examples, the dimensional transformation maycomprise the application of a matrix transformation. For example,digital signals corresponding to specific dimensions may undergo matrixmultiplication as part of the dimensional transformation. The matrixtransformation may be linear or substantially linear. The matrixtransformation may be a unitary or substantially unitary. That is, theinverse of the matrix transformation may be equal to or substantiallyequal to the complex conjugate transpose of the matrix transformation.In some examples, the linear operation based on matrix multiplicationmay be replaced by other forms of linear filtering. In some examples,the dimensional transformation may comprise using preliminary digitalsignals to determine corresponding transformed digital signals based oninformation stored in a database, such as a look-up table (LUT).

For the purposes of the following examples, the term “inversedimensional transformation” may be understood as an operation which isapplied to received digital signals, where the received digital signalscorrespond to dimensions of an optical signal received at a receiver.According to some examples, each multi-bit symbol may be represented byreceived digital signals corresponding to a plurality of firstdimensions of the optical signals, where the first dimensions may bedistributed over two or more distinct timeslots. Application of theinverse dimensional transformation may result in preliminary digitaldrive signal estimates, which correspond to a plurality of seconddimensions. According to some examples, the plurality of seconddimensions may be less than the plurality of first dimensions. In otherwords, the effect of the inverse dimensional transformation may be todecrease the number of dimensions over which each multi-bit symbol isrepresented, thereby resulting in preliminary digital drive signalestimates that represent each multi-bit symbol using fewer dimensionsthan the dimensions of the received optical signal that were used torepresent each multi-bit symbol. The decrease in “dimensionality” of themulti-bit symbols may facilitate soft-decoding at the receiver.

The inverse dimensional transformation may be implemented as one or moreserial steps, as one or more parallel steps, or as a combination of bothserial and parallel steps. In some examples, the inverse dimensionaltransformation may comprise the application of a matrix transformation.The matrix transformation may be linear or substantially linear. Thematrix transformation may be a unitary or substantially unitary. Anadvantage of using an inverse dimensional transformation that comprisesa unitary matrix is that application of such a matrix does not enhancenoise.

According to some examples, a dimensional transformation may be appliedto preliminary digital drive signals at a transmitter, therebygenerating transformed digital drive signals which are used to modulatean optical carrier to generate an optical signal. The optical signal maybe transmitted by the transmitter to a receiver. At the receiver, aninverse dimensional transformation may be applied to received digitalsignals, where the received digital signals correspond to dimensions ofa degraded version of the optical signal that was transmitted by thetransmitter. The inverse dimensional transformation may comprise anoperation that is substantially the inverse of a dimensionaltransformation applied at a transmitter. For example, where thedimensional transformation comprises the application of a first matrixtransformation, the inverse dimensional transformation may comprise theapplication of a second matrix transformation, where the second matrixtransformation is substantially the inverse of the first matrixtransformation. As a result of applying the inverse dimensionaltransformation to the received digital signals, preliminary digitaldrive signal estimates may be determined at the receiver. Thepreliminary digital drive signal estimates are estimates of thepreliminary digital drive signals to which the dimensionaltransformation was applied at the transmitter.

As will be described in the specific examples that follow, thedimensional transformation and the inverse dimensional transformationmay comprise additional operations, such as complex conjugate operationsthat are applied to a subset of signals, or signal interleaving.

When the range of noise levels of received signals are such that theycorrespond to the convex region of the curve that relates BER to linearNSR, such as the curve in FIG. 4, application of an inverse dimensionaltransformation, such as those described herein, may have an effect ofmaking the noise levels more uniform (i.e., by averaging the noiselevels, as described previously). However, when the range of noiselevels of the received signals are such that they correspond to theconcave region of the curve, application of the inverse dimensionaltransformation may be designed to have an effect of emphasizing thedistinction between the noise levels. For certain applications,enhancing the differences between noise levels may be advantageous. Inone example, multidimensional constellations that are non-prismatic maycover all four dimensions XI, XQ, YI, YQ within one or more timeslots. Adimensional transformation may be used to map streams of these symbolsinto purely X polarization and purely Y polarization dimensions acrosstwice as many timeslots. PDL may produce unequal noise variances onthese streams. When the range of noise levels of the received signalsare such that they correspond to the concave region of the curve,unequal noise variances may be better handled FEC. Therefore, in suchcircumstances, it may be of interest to emphasize the inequality usingthe inverse dimensional transformation.

Referring to FIG. 2, the signal processing performed at the transmit DSP214 may comprise applying a dimensional transformation to preliminarydigital drive signals, which may be denoted by Ŝ_(XI), Ŝ_(XQ), Ŝ_(YI),Ŝ_(YQ). For simplicity, Ŝ_(X) may be used throughout this document todenote the combination of Ŝ_(XI) and Ŝ_(XQ), while Ŝ_(Y) may be usedthroughout this document to denote the combination of Ŝ_(YI) and Ŝ_(YQ).For simplicity of this description, the proposed technology is describedin terms of modifications applied to traditional systems and methods.Therefore, in FIG. 2, the preliminary digital drive signals are thedigital drive signals determined by the transmitter, based on a specificmodulation scheme, to be used in the commonly understood case formodulating orthogonal polarizations of an optical carrier in order torepresent multi-bit symbols. That is, the preliminary digital drivesignals Ŝ_(X), Ŝ_(Y), are designed for modulating a plurality ofdimensions of the optical carrier in order to represent digitalinformation according to a specific modulation scheme. However, theproposed technology need not be implemented as a change to knownmethods, and thus the preliminary digital drive signals could be anymodulation of a plurality of mathematical dimensions. The preliminarydigital drive signals may be most simply represented by one physicaldigital integer per dimension, per timeslot. However, a generallyequivalent function may be obtained with other representations, or bybeing part of a mathematical operation beyond what is described in theseexamples. The application of the dimensional transformation to thepreliminary digital drive signals Ŝ_(X), Ŝ_(Y) may generate transformeddigital drive signals. The transformed digital drive signals may bedenoted by S_(X), S_(Y), respectively, where S_(X) is used throughoutthis document to denote the combination of S_(XI) and S_(XQ), and S_(Y)is used throughout this document to denote the combination of S_(YI) andS_(YQ). As will be described further with respect to specific examples,application of the dimensional transformation to a plurality ofpreliminary digital drive signals may result in a plurality oftransformed digital drive signals, where each transformed digital drivesignal is to be used in the modulation of a respective one of aplurality of dimensions of the optical carrier, and where the dimensionsare distributed over two or more distinct timeslots. In some examples,the plurality of preliminary digital drive signals to which thedimensional transformation is applied may also be representative of twoor more distinct timeslots.

Referring to FIG. 3, the signal processing performed at the receive DSP314 may comprise applying an inverse dimensional transformation toreceived digital signals, which may be denoted by R_(XI), R_(XQ),R_(YI), R_(YQ). For simplicity, R_(X) may be used throughout thisdocument to denote the combination of R_(XI) and R_(XQ), while R_(Y) maybe used throughout this document to denote the combination of R_(YI) andR_(YQ). The application of the inverse dimensional transformation toreceived digital signals R_(X), R_(Y) may generate digital signals{circumflex over (R)}_(X), {circumflex over (R)}_(Y), respectively,where {circumflex over (R)}_(X) is used throughout this document todenote the combination of {circumflex over (R)}_(XI) and {circumflexover (R)}_(XQ), and {circumflex over (R)}_(Y) is used throughout thisdocument to denote the combination of {circumflex over (R)}_(YI) and{circumflex over (R)}_(YQ). The digital signals {circumflex over(R)}_(X), {circumflex over (R)}_(Y) correspond to estimates ofpreliminary digital drive signals Ŝ_(X), Ŝ_(Y), respectively. Thecarrier recovery process 313 may be applied to the preliminary digitaldrive signal estimates {circumflex over (R)}_(X), {circumflex over(R)}_(Y). As will be described further with respect to specificexamples, the inverse dimensional transformation may be applied to aplurality of received digital signals, where each received digitalsignal is representative of a respective one of a plurality ofdimensions of a received optical signal, and where the dimensions aredistributed over two or more distinct timeslots. In some examples, thepreliminary digital drive signal estimates that result from the inversedimensional transformation may also be representative of dimensions thatare distributed over two or more distinct timeslots.

The application of the dimensional transformation at the transmitter andthe inverse dimensional transformation at the receiver differs from thedisclosures of Khandani et al. in U.S. Pat. Nos. 8,718,491 and9,602,207, in which a transmit Jones rotation matrix is applied at thetransmitter, and a noise whitening matrix is applied at the receiver.The noise whitening matrix is not the inverse of the Jones rotationmatrix. Furthermore, the dimensional transformation disclosed herein maybe used to average the noise across polarizations, without tracking thechanging optical line.

In contrast to the gold and silver codes described by Mumtaz et al., theapplication of the dimensional transformation and inverse dimensionaltransformation, as described herein, does not require complex circuitryto implement. Multiplication with a unitary matrix involves simple,inexpensive computations relative to those needed to implement gold andsilver codes. Accordingly, a dimensional transformation may provide analternative to gold and silver codes that is less costly in terms ofheat production and power use.

In “Filtering-tolerant transmission by the Walsh-Hadamard transform forsuper-channel beyond 100 Gb/s,” Optical Society of America, 2015,Shibahara et al. describe a method for improving super-channelperformance by dispersing optical filtering distortions over allsubcarriers of a super-channel. The method involves applying aWalsh-Hadamard transform (WHT) to the subcarriers, where each subcarriercorresponds to a different wavelength.

In “Twin-Wave-Based Optical Transmission with Enhanced Linear andNonlinear Performances,”, Journal of Lightwave Technology, Vol. 33,Issue 5, pp. 1037-1043 (2015), Liu describes a method for converting abinary phase-shift keying (BPSK) signal to a “Twin-Wave” QPSK signal,with conjugate phase properties. Liu's method involves matrixmultiplication using a unitary matrix. However, Liu's method does notinvolve received digital signals corresponding to first dimensions of anoptical signal that are representative of a single multi-bit symbol,where the first dimensions are distributed over two or more distincttimeslots. That is, Liu's method does not involve the application of atime-memory or inter-time transformation. Liu's method involves BPSKwhich encodes one bit per symbol.

In “A Pragmatic Approach to Trellis-Coded Modulation,” IEEECommunications Magazine, Vol. 27, Issue 7, pp. 11-19 (1989), Viterbi etal. describe techniques for trellis or convolution coding, in which theeffect of a symbol may be distributed across multiple timeslots. Inorder to decode a bitstream that has been encoded using a trellis code,a Viterbi decoder may be used. The decoding of a trellis-encodedbitstream does not involve any inverse dimensional transformation thathas the effect of reducing the dimensionality of symbols.

U.S. Pat. No. 3,388,330 to Kretzmer et al. describes a partial responsemultilevel data system in which channel response to a single symbolextends over more than one symbol interval. Kretzmer et al. do notdescribe any inverse dimensional transformation that has the effect ofreducing the dimensionality of symbols.

FIG. 7 illustrates an example method 700 for implementing a dimensionaltransformation at a transmitter, such as the transmitter 200. The method700 may be implemented by a DSP of the transmitter, such as the transmitDSP 214.

At 702, based on a specific modulation scheme, the transmitter maydetermine preliminary digital drive signals to be used for modulatingdimensions of an optical carrier in order to represent multi-bit symbolsof a symbol stream. Each multi-bit symbol may be represented bypreliminary digital drive signals that correspond to a plurality ofdimensions, where the dimensions comprise a specific combination of thedimensions XI, XQ, YI, YQ at a single timeslot. For simplicity, thedimensions over which each multi-bit symbol is represented using thepreliminary digital drive signals are herein denoted as “seconddimensions.” The preliminary digital drive signals at a timeslot, t, maybe denoted Ŝ_(X)(t), Ŝ_(Y)(t).

At 704, the transmitter may determine transformed digital drive signalsbased on a dimensional transformation and the preliminary digital drivesignals determined at 702. In one example, the transmitter may generatethe transformed digital drive signals by applying the dimensionaltransformation directly to the preliminary digital drive signalsdetermined at 702. In another example, the transmitter may generate thetransformed digital drive signals by applying the dimensionaltransformation to digital signals that are based on the preliminarydigital drive signals determined at 702. In another example, thetransmitter may determine the transformed digital drive signals using aLUT corresponding to the dimensional transformation.

Application of the dimensional transformation may result in transformeddigital drive signals that are designed to modulate the optical carriersuch that each multi-bit symbol is represented by a plurality ofdimensions of the optical carrier, which are denoted herein as “firstdimensions” to distinguish them from the dimensions over which eachmulti-bit symbol is represented using the preliminary digital drivesignals. The first dimensions differ from the second dimensions. Thefirst dimensions comprise a specific combination of the dimensions XI,XQ, YI, YQ at two or more distinct timeslots. According to someexamples, the plurality of second dimensions is less than the pluralityof first dimensions. Given preliminary digital drive signals at atimeslot, t, denoted Ŝ_(X)(t), Ŝ_(Y)(t), the transformed digital drivesignals at the same timeslot, t, may be denoted S_(X)(t), S_(Y)(t),respectively.

At 706, the transmitter may generate a modulated optical signal usingthe transformed digital drive signals that were determined at 704. Forexample, as described with respect to FIG. 2, the generation of themodulated optical signal may be achieved by converting the transformeddigital drive signals S_(X)(t), S_(Y)(t), into respective analog drivesignals, driving electrical-to-optical modulators with the analog drivesignals to generate modulated polarized optical signals, and combiningthe modulated polarized optical signals to form an optical signal, suchas the optical signal 260. Instead of the modulated optical signalhaving been generated using the preliminary digital drive signals, whichwere designed to represent each multi-bit symbol using a plurality ofsecond dimensions, the modulated optical signal is generated using thetransformed digital drive signals, which are designed to represent eachmulti-bit symbol using a plurality of first dimensions, where the firstdimensions are distributed over two or more distinct timeslots.

At 708, the transmitter may transmit the modulated optical signal over acommunications channel. As a result of the modulation having beenperformed using the transformed digital drive signals, each multi-bitsymbol may be represented using first dimensions of the optical signal,where the first dimensions are distributed over two or more distincttimeslots.

FIG. 8 illustrates an example method 800 for implementing an inversedimensional transformation at a receiver, such as the receiver 300. Themethod 800 may be implemented by a DSP of the receiver, such as thereceive DSP 314.

At 802, the receiver may receive an optical signal. The received opticalsignal may be representative of a stream of multi-bit symbols. Accordingto some examples, received optical signal may comprise a degradedversion of a modulated optical signal generated at a transmitteraccording to the method 700. That is, the received optical signal mayhave been generated by modulating a plurality of first dimensions of anoptical carrier to represent each multi-bit symbol. The first dimensionsmay comprise a specific combination of the dimensions XI, XQ, YI, YQ attwo or more timeslots.

At 804, the receiver may determine digital signals corresponding todimensions of the received optical signal. For example, as describedwith respect to FIG. 3, a received optical signal, such as the signal360, may be split into orthogonally-polarized components using apolarizing beam splitter. An optical hybrid may process the componentswith respect to an optical signal, and photodetectors may convert theoutputs of the optical hybrid to analog signals, which may be convertedto received digital signals. At a particular timeslot, t, the receiveddigital signals may be denoted by R_(X)(t), R_(Y)(t).

At 806, the receiver may determine preliminary digital drive signalestimates based on an inverse dimensional transformation and thereceived digital signals determined at 804. In one example, the receivermay generate the preliminary digital drive signal estimates by applyingthe inverse dimensional transformation directly to the received digitalsignals determined at 804. In another example, the receiver may generatethe preliminary digital drive signal estimates by applying the inversedimensional transformation to digital signals that are based on thereceived digital signals determined at 804.

Application of the inverse dimensional transformation results in eachmulti-bit symbol being represented by preliminary digital drive signalestimates that correspond to a plurality of dimensions, which aredenoted herein as “second dimensions” to distinguish them from thedimensions over which each multi-bit symbol is represented using thereceived digital signals. The second dimensions may correspond to thesecond dimensions described with respect to the method 700. The seconddimensions differ from the first dimensions. The second dimensionscomprise a specific combination of the dimensions XI, XQ, YI, YQ at asingle timeslot. According to some examples, the plurality of seconddimensions is less than the plurality of first dimensions. Givenreceived digital signals at a timeslot, t, denoted R_(X)(t), R_(Y)(t),the preliminary digital drive signal estimates at the same timeslot, t,may be denoted {circumflex over (R)}_(X)(t), {circumflex over(R)}_(Y)(t), respectively. Where the inverse dimensional transformationis substantially the inverse of a dimensional transformation that wasapplied at a transmitter at 704, the digital signals {circumflex over(R)}_(X)(t), {circumflex over (R)}_(Y)(t) may be estimates of thepreliminary digital drive signals Ŝ_(X)(t), Ŝ_(Y)(t), respectively, thatwere determined at 702.

At 808, the receiver may determine estimates of multi-bit symbols usingthe preliminary digital drive signal estimates determined at 806. Forexample, this determination may include applying the carrier recoveryprocess 313, as described with respect to FIG. 3, to the digital signals{circumflex over (R)}_(X)(t), {circumflex over (R)}_(Y)(t) generated at804. Each symbol estimate determined at 808 may subsequently undergosymbol-to-bit mapping, such as that denoted by 310 in FIG. 3, in orderto recover corresponding bit estimates. Where the symbols are comprisedof FEC-encoded bits, the bit estimates may subsequently undergo FECdecoding, such at that denoted by 306 in FIG. 3, thereby generatingcorrected client data bits.

The remainder of this document provides example techniques forimplementing a dimensional transformation at a transmitter and acorresponding inverse dimensional transformation at a receiver. In thefollowing examples, the dimensional transformation is applied by a DSPof a transmitter, such as the transmit DSP 214 of the transmitter 200.The inverse dimensional transformation is applied by a DSP of areceiver, such as the receiver DSP 314 of the receiver 300.

Application of the dimensional transformations and corresponding inversedimensional transformations described in the following examples may beused to average signal degradations across a plurality of signaldimensions, including degradations caused by one or more of noise,nonlinear effects, PDL, and analog imperfections.

According to some examples, matched filtering may be applied at thetransmitter and receiver, in order to achieve low noise levels.Substantially zero inter-symbol interference may be achieved, forexample, using a matched filter selected from the raised cosine family.

According to some examples, an adaptive equalization circuit may beemployed at the receiver to correct for PMD, PDL, and other linearvariations. This equalization can be performed in the time domain, orthe frequency domain, or both, or with other transformations. Commonmethods for controlling this equalization include recursive leastsquares (RLS) equalization, constant modulus algorithm (CMA)equalization, least means squares (LMS) equalization, and decisionfeedback equalization (DFE). LMS equalization may provide anadvantageous compromise between complexity and performance. An LMScircuit may result in noise correlation for symbols that are within acertain number, N, of integer timeslots from each other and/or forsymbols over different polarizations. As previously noted, theapplication of a dimensional transformation and an inverse dimensionalmay involve sets of signals that are representative of at least a firsttimeslot and a second timeslot, where the timeslots are separated by aninteger number, T. Where an LMS circuit is used in such examples, it maybe of interest to select T to be greater than the number, M, oftimeslots over which the LMS circuit generates noise correlation and/oruneven noise boosting. In this manner, noise averaging achieved by thedimensional transformation (and inverse dimensional transformation) maynot be impeded as a result of the noise correlation caused by the LMScircuit. Furthermore, the dimensional transformation may be applied overdimensions with different noise levels to ensure an averaged noise levelover different dimensions.

Example 1

FIG. 9 is a schematic diagram illustrating the implementation of adimensional transformation at a transmitter according to a firstexample. In this example, the dimensional transformation comprises amatrix transformation H₁ provided in Eq. 1:

$\begin{matrix}{H_{1}\overset{\Delta}{=}\begin{pmatrix}1 & 0 & 0 & 1 \\0 & 1 & 1 & 0 \\0 & {- 1} & 1 & 0 \\{- 1} & 0 & 0 & 1\end{pmatrix}} & (1)\end{matrix}$

The matrix transformation H₁ may be applied to the preliminary digitaldrive signals Ŝ_(X)(t−T), Ŝ_(Y)(t−T), Ŝ_(X)(t), Ŝ_(Y)(t) to generatesignals S_(X)(t−T), S_(Y)(t−T), S_(X)*(t), S_(Y)*(t), respectively,where t−T denotes a first integer timeslot, and t denotes a secondinteger timeslot. This is shown in Eq. 2.

$\begin{matrix}{\begin{pmatrix}{S_{X}\left( {t - T} \right)} \\{S_{Y}\left( {t - T} \right)} \\{S_{X}^{*}(t)} \\{S_{Y}^{*}(t)}\end{pmatrix} = {{H_{1}\begin{pmatrix}{{\hat{S}}_{X}\left( {t - T} \right)} \\{{\hat{S}}_{Y}\left( {t - T} \right)} \\{{\hat{S}}_{X}(t)} \\{{\hat{S}}_{Y}(t)}\end{pmatrix}} = {\begin{pmatrix}1 & 0 & 0 & 1 \\0 & 1 & 1 & 0 \\0 & {- 1} & 1 & 0 \\{- 1} & 0 & 0 & 1\end{pmatrix}\begin{pmatrix}{{\hat{S}}_{X}\left( {t - T} \right)} \\{{\hat{S}}_{Y}\left( {t - T} \right)} \\{{\hat{S}}_{X}(t)} \\{{\hat{S}}_{Y}(t)}\end{pmatrix}}}} & (2)\end{matrix}$

The signals S_(X)(t−T) and S_(Y)(t−T) denote the transformed digitaldrive signals at the first timeslot, t−T. The transformed digital drivesignals at the second timeslot, t, that is S_(X)(t) and S_(Y)(t), may bedetermined by taking the complex conjugate of the signals S_(X)*(t) andS_(Y)*(t), respectively. Since the complex conjugate operation is onlyapplied to the signals at the second timeslot t, and not the signals atthe first timeslot, t−T, the complex conjugate operation may be referredto as a “partial complex conjugate”.

As illustrated in FIG. 9, the combination of the matrix transformationH₁ followed by the partial complex conjugate is denoted by dimensionaltransformation 902. The dimensional transformation 902 is applied topreliminary digital drive signals 901 to generate transformed digitaldrive signals 903. The transformed digital drive signals 903 may undergoadditional processing before being transformed to analog drive signals.For example, a FFT 904 may be applied to the transformed digital drivesignals 903, thereby producing frequency-domain signals 905, which maysubsequently undergo frequency-domain processing 906 to produceprocessed frequency-domain signals 907. The frequency-domain processing906 may include the application of a matched filter. The processedfrequency-domain signals 907 may be converted to correspondingtime-domain signals 909 by an IFFT 908.

FIG. 10 is a schematic diagram illustrating example details forimplementing the dimensional transformation according to the firstexample. That is, the dimensional transformation 902 described withrespect to FIG. 9 may be implemented using the operations performed inFIG. 10.

Given preliminary digital drive signals Ŝ_(X)(t), Ŝ_(Y)(t), applicationof a delay of T timeslots results in preliminary digital drive signalsŜ_(X)(t−T), Ŝ_(Y)(t−T), respectively. This delay is denoted by box 1002.

As denoted by box 1004, the preliminary digital drive signals Ŝ_(X)(t),Ŝ_(Y)(t), Ŝ_(X)(t−T), Ŝ_(Y)(t−T) are partitioned into pairs, such thatsignal Ŝ_(X)(t) is paired with the signal Ŝ_(Y)(t−T), while the signalŜ_(Y)(t) is paired with the signal Ŝ_(X)(t−T).

As denoted by box 1006, the preliminary digital drive signals Ŝ_(X)(t−T)and Ŝ_(Y)(t) undergo a 45-degree rotation, which results in the signalsS_(X)(t−T) and S_(Y)*(t), respectively.

As denoted by box 1008, the preliminary digital drive signals Ŝ_(Y)(t−T)and Ŝ_(X)(t) may also undergo a 45-degree rotation, which results in thesignals S_(Y)(t−T) and S_(X)*(t), respectively.

The signals S_(X)(t−T) and S_(Y)(t−T) are the transformed digital drivesignals at the first timeslot, t−T. The signals S_(Y)*(t) and S_(X)*(t)may undergo a complex conjugate operation, denoted by box 1010, togenerate the signals S_(X)(t) and S_(Y)(t), respectively, which are thetransformed digital drive signals at the second timeslot, t.

Accordingly, the operations performed in FIG. 10 may be used to achievethe dimensional transformation 902 described with respect to FIG. 9, bytransforming preliminary digital drive signals 901 at a specifictimeslot, t (e.g., Ŝ_(X)(t), Ŝ_(Y)(t) as illustrated in FIG. 10) intotransformed digital drive signals 903 at the same specific timeslot, t(e.g., S_(X)(t), S_(Y)(t) as illustrated in FIG. 10). The operations inFIG. 10 demonstrate merely one example of how the dimensionaltransformation 902 may be implemented.

FIG. 11 is a schematic diagram illustrating the implementation of aninverse dimensional transformation at a receiver according to the firstexample. In this example, the inverse dimensional transformationcomprises an inverse matrix transformation H₁ ⁻¹ provided in Eq. 3:

$\begin{matrix}{H_{1}^{- 1}\overset{\Delta}{=}\begin{pmatrix}1 & 0 & 0 & {- 1} \\0 & 1 & {- 1} & 0 \\0 & 1 & 1 & 0 \\1 & 0 & 0 & 1\end{pmatrix}} & (3)\end{matrix}$

Application of the inverse matrix transformation H₁ ⁻¹ at the receiveris included as part of inverse dimensional transformation 1108, which isthe inverse of the dimensional transformation 902 that was applied atthe transmitter. Received digital signals R_(X)(t−T), R_(Y)(t−T),R_(X)(t), R_(Y)(t) may undergo a partial complex conjugate operation toproduce signals R_(X)(t−T), R_(Y)(t−T), R_(X)*(t), R_(Y)*(t), where t−Tdenotes a first integer timeslot, and t denotes a second integertimeslot. The inverse matrix transformation H₁ ⁻¹ may then be applied tothe signals R_(X)(t−T), R_(Y)(t−T), R_(X)*(t), R_(Y)*(t) to generatesignals {circumflex over (R)}_(X)(t−T), {circumflex over (R)}_(Y)(t−T),{circumflex over (R)}_(X)(t), {circumflex over (R)}_(Y)(t),respectively. This is shown in Eq. 4.

$\begin{matrix}{\begin{pmatrix}{{\hat{R}}_{X}\left( {t - T} \right)} \\{{\hat{R}}_{Y}\left( {t - T} \right)} \\{{\hat{R}}_{X}(t)} \\{{\hat{R}}_{Y}(t)}\end{pmatrix} = {{H_{1}^{- 1}\begin{pmatrix}{R_{X}\left( {t - T} \right)} \\{R_{Y}\left( {t - T} \right)} \\{R_{X}^{*}(t)} \\{R_{Y}^{*}(t)}\end{pmatrix}} = {\begin{pmatrix}1 & 0 & 0 & {- 1} \\0 & 1 & {- 1} & 0 \\0 & 1 & 1 & 0 \\1 & 0 & 0 & 1\end{pmatrix}\begin{pmatrix}{R_{X}\left( {t - T} \right)} \\{R_{Y}\left( {t - T} \right)} \\{R_{X}^{*}(t)} \\{R_{Y}^{*}(t)}\end{pmatrix}}}} & (4)\end{matrix}$

The signals {circumflex over (R)}_(X)(t−T) and {circumflex over(R)}_(Y)(t−T) denote preliminary digital drive signal estimates at thefirst timeslot, t−T, while the signals {circumflex over (R)}_(X)(t) and{circumflex over (R)}_(Y)(t) denote preliminary digital drive signalestimates at the second timeslot, t. Referring to FIG. 11, the inversedimensional transformation 1104 is applied to received digital signals1103 to produce preliminary digital drive signal estimates 1105. Thepreliminary digital drive signal estimates 1105 may subsequently undergocarrier recovery 1106, to generate symbol estimates 1107, followed bysymbol-to-bit demapping 1108, to generate bit estimates 1109.

Prior to undergoing the inverse dimensional transformation 1104, thereceived digital signals 1103 may have undergone additional processing.For example, the received digital signals 1103 may result from applyingadaptive equalization 1102 to digital signals 1101, in order tocompensate for channel linear impairments, such as PMD and PDL. Theadaptive equalization 1102 may be implemented using a variety ofalgorithms, such as LMS, CMA, RLS, and DFE. The adaptive equalization1102 may be applied in either the time domain or the frequency domain.In one example, a FFT may be applied to digital signals generated fromanalog-to-digital conversion, thereby producing frequency-domainsignals, which may be processed using adaptive equalization in thefrequency domain. The processed frequency-domain signals may then beconverted to corresponding time-domain signals by an IFFT.

Parameters used for the adaptive equalization 1102 may be updated aschannel linear distortions evolve over time. In some examples, theparameters may be updated based on error values determined from thedifference between an ideal target signal and a received signal. Inother examples, the parameters may be updated based on a calculation ofthe value of the target signal. In some examples, the preliminarydigital drive signal estimates 1105 may undergo an equalizererror/target calculation 1110 to generate values 1111. In some examples,the calculation 1110 may involve a LUT. Dimensional transformation 1112,which is identical to the dimensional transformation 902, may be appliedto the values 1111 to generate transformed values 1113, which are usedto guide parameters used for the adaptive equalization 1102. As denotedby the dashed-line, in some examples, the equalizer error/targetcalculation 1110 may be applied to the symbol estimates 1107 generatedby the carrier recovery 1106, instead of the preliminary digital drivesignal estimates 1105 generated by the inverse dimensionaltransformation 1104.

FIG. 12 is a schematic diagram illustrating example details forimplementing the inverse dimensional according to the first example.That is, the inverse dimensional transformation 1104 described withrespect to FIG. 11 may be implemented using the operations performed inFIG. 12.

Given received digital signals R_(X)(t), R_(Y)(t), application of adelay of T timeslots results in received digital signals R_(X)(t−T),R_(Y)(t−T), respectively. This delay is denoted by box 1202.

As denoted by box 1204, the received digital signals R_(X)(t), R_(Y)(t),R_(X)(t−T), R_(Y)(t−T) are partitioned into pairs, such that signalR_(X)(t−T) is paired with the signal R_(Y)(t), while the signal R_(X)(t)is paired with the signal R_(Y)(t−T).

The signals R_(X)(t) and R_(Y)(t) may undergo a complex conjugateoperation, denoted by box 1206, to generate the signals R_(X)*(t) andR_(Y)*(t), respectively.

As denoted by box 1208, the signals R_(X)(t−T) and R_(Y)*(t) may undergoa 45-degree rotation, which results in the signals {circumflex over(R)}_(X)(t−T) and {circumflex over (R)}_(Y)(t), respectively.

As denoted by box 1210, the signals R_(X)*(t) and R_(Y)(t−T) may alsoundergo a 45-degree rotation, which results in the signals {circumflexover (R)}_(X)(t) and {circumflex over (R)}_(Y)(t−T), respectively.

The signals {circumflex over (R)}_(X)(t−T) and {circumflex over(R)}_(Y)(t−T) are the preliminary digital drive signal estimates at thefirst timeslot, t−T. The signals {circumflex over (R)}_(X)(t) and{circumflex over (R)}_(Y)(t) are the preliminary digital drive signalestimates at the second timeslot, t.

Accordingly, the operations performed in FIG. 12 may be used to achievethe inverse dimensional transformation 1104 described with respect toFIG. 11, by transforming received digital signals 1103 at a specifictimeslot, t (e.g., R_(X)(t), R_(Y)(t) as illustrated in FIG. 12) intopreliminary digital drive signal estimates 1105 at the same specifictimeslot, t (e.g., {circumflex over (R)}_(X)(t), {circumflex over(R)}_(Y)(t) as illustrated in FIG. 12). The operations in FIG. 12demonstrate merely one example of how the inverse dimensionaltransformation 1104 may be implemented.

It may be demonstrated computationally that the aggregate NSR of thepreliminary digital drive signal estimates 1105 is identical to theaggregate NSR of the received digital signals 1103. That is, the inversedimensional transformation 1104 does not alter the average NSR. Instead,the inverse dimensional transformation 1104 redistributes or averagesthe noise or other degradations across signal dimensions.

Example 2

According to a second example, a dimensional transformation comprises amatrix transformation H₂ provided in Eq. 5:

$\begin{matrix}{H_{2}\overset{\Delta}{=}\begin{pmatrix}1 & 1 \\{- 1} & 1\end{pmatrix}} & (5)\end{matrix}$

The matrix transformation H₂ may be applied to the preliminary digitaldrive signals Ŝ_(X)(t) and Ŝ_(Y)(t) to generate transformed digitaldrive signals S_(X)(t−T) and S_(Y)(t), respectively, where t−T denotes afirst integer timeslot, and t denotes a second integer timeslot. This isshown in Eq. 6.

$\begin{matrix}{\begin{pmatrix}{S_{X}\left( {t - T} \right)} \\{S_{Y}(t)}\end{pmatrix} = {{H_{2}\begin{pmatrix}{{\hat{S}}_{X}(t)} \\{{\hat{S}}_{Y}(t)}\end{pmatrix}} = {\begin{pmatrix}1 & 1 \\{- 1} & 1\end{pmatrix}\begin{pmatrix}{{\hat{S}}_{X}(t)} \\{{\hat{S}}_{Y}(t)}\end{pmatrix}}}} & (6)\end{matrix}$

The signal S_(X)(t−T) denotes the transformed digital drive signal inthe X polarization at the first timeslot, t−T, while the signal S_(Y)(t)denotes the transformed digital drive signal in the Y polarization atthe second timeslot, t.

FIG. 13 is a schematic diagram illustrating example details forimplementing a dimensional transformation at a transmitter according tothe second example.

As denoted by box 1302, the preliminary digital drive signals Ŝ_(X)(t)and Ŝ_(Y)(t) undergo a 45-degree rotation, which results in the signalsS_(X)(t) and S_(Y)(t), respectively.

As denoted by box 1304, a FFT is applied to the signals S_(X)(t) andS_(Y)(t), thereby producing frequency-domain signals S_(X)(f) andS_(Y)(f), respectively.

As denoted by box 1306, the signals S_(X)(f) and S_(Y)(f) may undergofrequency-domain processing to produce signals S_(X)′(f) and S_(Y)′(f),respectively. The processing 1306 may include the application of a delayof T timeslots to signal S_(X) relative to the signal S_(Y).

The processed frequency-domain signals S_(X)′(f) and S_(Y)′(f) may beconverted by an IFFT 1308 to corresponding time-domain signals, denotedS_(X)(t−T) and S_(Y)(t), respectively.

Accordingly, the operations performed in FIG. 13 may be used to achievethe dimensional transformation denoted by Eq. 6, by transformingpreliminary digital drive signals Ŝ_(X)(t), Ŝ_(Y)(t) into transformeddigital drive signals S_(X)(t−T), S_(Y)(t). The operations in FIG. 13demonstrate merely one example of how the dimensional transformationdenoted by Eq. 6 may be implemented.

According to the second example, the inverse dimensional transformationcomprises an inverse matrix transformation H₂ ⁻¹ provided in Eq. 7:

$\begin{matrix}{H_{2}^{- 1}\overset{\Delta}{=}\begin{pmatrix}1 & {- 1} \\1 & 1\end{pmatrix}} & (7)\end{matrix}$

The inverse matrix transformation H₂ ⁻¹ may be applied to receiveddigital signals R_(X)(t) and R_(Y)(t−T) to generate preliminary digitaldrive signal estimates {circumflex over (R)}_(X)(t) and {circumflex over(R)}_(Y)(t), respectively, where t denotes a first integer timeslot, andt−T denotes a second integer timeslot. This is shown in Eq. 8.

$\begin{matrix}{\begin{pmatrix}{{\hat{R}}_{X}(t)} \\{{\hat{R}}_{Y}(t)}\end{pmatrix} = {{H_{2}^{- 1}\begin{pmatrix}{R_{X}\left( {t - T} \right)} \\{R_{Y}(t)}\end{pmatrix}} = {\begin{pmatrix}1 & {- 1} \\1 & 1\end{pmatrix}\begin{pmatrix}{R_{X}\left( {t - T} \right)} \\{R_{Y}(t)}\end{pmatrix}}}} & (8)\end{matrix}$

FIG. 14 is a schematic diagram illustrating the implementation of aninverse dimensional transformation at a receiver according to the secondexample.

Given received digital signals R_(X)(t) and R_(Y)(t) at a secondtimeslot, t, such as the signals 1107 described with respect to FIG. 11,a delay of T timeslots may be applied to the received digital signalR_(X)(t), thereby producing R_(X)(t−T) at a first timeslot t−T. Thisdelay is denoted by box 1402.

As denoted by box 1404, the signals R_(X)(t−T) and R_(Y)(t) may undergoa 45-degree rotation, which results in the signals {circumflex over(R)}_(X)(t) and {circumflex over (R)}_(Y)(t), respectively. The signals{circumflex over (R)}_(X)(t) and {circumflex over (R)}_(Y)(t) are thepreliminary digital drive signal estimates at the second timeslot, t.

Accordingly, the operations performed in FIG. 14 may be used to achievethe inverse dimensional transformation denoted by Eq. 8, by transformingreceived digital signals R_(X)(t−T), R_(Y)(t) into preliminary digitaldrive signal estimates {circumflex over (R)}_(X)(t), {circumflex over(R)}_(Y)(t), respectively. The operations in FIG. 14 may be implementedin place of the inverse dimensional transformation 1108 in FIG. 11. Inthis case, the operations illustrated in FIG. 13 would be implemented inplace of the dimensional transformation 902 in FIG. 9.

The dimensional transformation of Example 2 (see Eq. 6 and FIG. 13) is alinear time-invariant operation. Accordingly, it may be inverted usingan adaptive equalizer circuit, which may be implemented, for example, inthe receive DSP 314 as part of the channel impairment compensation.Indeed, the dimensional transformation of Example 2 may be thought of asan example of a time-invariant linear transformation such as the oneapplied by the channel itself, but which is intentionally applied at thetransmitter. Thus, the adaptive equalization performed at the receivermay be able to invert the dimensional transformation together the withchannel linear impairments. In contrast, the dimensional transformationof Example 1 (see Eq. 2 and FIG. 10) is a time-variant transformation,since it involves partitioning time samples into pairs. Accordingly, achannel equalizer at the receiver may not be capable of inverting suchan operation.

As a result of an I/Q power imbalance or timing skew, the noise powermay differ between dimensions XI, XQ, YI, and YQ at a given timeslot.Examples 3 and 4 below describe modified versions of the matrixtransformations H₁ and H₂, respectively, that may average outimpairments which affect the I and Q components of the X and Ypolarizations differently.

Example 3

According to a third example, a dimensional transformation comprising amodified version of the matrix transformation H₁ of the first examplemay be implemented at a transmitter. In this third example, the matrixtransformation, denoted H₃, is provided by Eq. 9:

$\begin{matrix}{H_{3}\overset{\Delta}{=}{e^{\frac{j\; \pi}{4}}\begin{pmatrix}1 & 0 & 0 & {1j} \\0 & 1 & {1j} & 0 \\0 & {1j} & 1 & 0 \\{1j} & 0 & 0 & 1\end{pmatrix}}} & (9)\end{matrix}$

The matrix transformation H₃ may be used in place of the matrixtransformation H₁ in Eq. 2, thereby resulting in Eq. 10:

$\begin{matrix}{\begin{pmatrix}{S_{X}\left( {t - T} \right)} \\{S_{Y}\left( {t - T} \right)} \\{S_{X}^{*}(t)} \\{S_{Y}^{*}(t)}\end{pmatrix} = {{H_{3}\begin{pmatrix}{{\hat{S}}_{X}\left( {t - T} \right)} \\{{\hat{S}}_{Y}\left( {t - T} \right)} \\{{\hat{S}}_{X}(t)} \\{{\hat{S}}_{Y}(t)}\end{pmatrix}} = {{e^{\frac{j\; \pi}{4}}\begin{pmatrix}1 & 0 & 0 & {1j} \\0 & 1 & {1j} & 0 \\0 & {1j} & 1 & 0 \\{1j} & 0 & 0 & 1\end{pmatrix}}\begin{pmatrix}{{\hat{S}}_{X}\left( {t - T} \right)} \\{{\hat{S}}_{Y}\left( {t - T} \right)} \\{{\hat{S}}_{X}(t)} \\{{\hat{S}}_{Y}(t)}\end{pmatrix}}}} & (10)\end{matrix}$

As described with respect to Eq. 2, the signals S_(X)(t−T) andS_(Y)(t−T) denote the transformed digital drive signals at the firsttimeslot, t−T. The transformed digital drive signals at the secondtimeslot, t, that is S_(X)(t) and S_(Y)(t), may be determined by takingthe complex conjugate of the signals S_(X)*(t) and S_(Y)*(t),respectively.

According to the third example, the inverse dimensional transformationcomprises an inverse matrix transformation H₃ ⁻¹ that is provided in Eq.11:

$\begin{matrix}{H_{3}^{- 1}\overset{\Delta}{=}{e^{- \frac{j\; \pi}{4}}\begin{pmatrix}1 & 0 & 0 & {{- 1}j} \\0 & 1 & {{- 1}j} & 0 \\0 & {{- 1}j} & 1 & 0 \\{{- 1}j} & 0 & 0 & 1\end{pmatrix}}} & (11)\end{matrix}$

The inverse matrix transformation H₃ ⁻¹ may be used in place of theinverse matrix transformation H₁ ⁻¹ in Eq. 4, thereby resulting in Eq.12:

$\begin{matrix}{\begin{pmatrix}{{\hat{R}}_{X}\left( {t - T} \right)} \\{{\hat{R}}_{Y}\left( {t - T} \right)} \\{{\hat{R}}_{X}(t)} \\{{\hat{R}}_{Y}(t)}\end{pmatrix} = {{H_{3}^{- 1}\begin{pmatrix}{R_{X}\left( {t - T} \right)} \\{R_{Y}\left( {t - T} \right)} \\{R_{X}^{*}(t)} \\{R_{Y}^{*}(t)}\end{pmatrix}} = {{e^{- \frac{j\; \pi}{4}}\begin{pmatrix}1 & 0 & 0 & {{- 1}j} \\0 & 1 & {{- 1}j} & 0 \\0 & {{- 1}j} & 1 & 0 \\{{- 1}j} & 0 & 0 & 1\end{pmatrix}}\begin{pmatrix}{R_{X}\left( {t - T} \right)} \\{R_{Y}\left( {t - T} \right)} \\{R_{X}^{*}(t)} \\{R_{Y}^{*}(t)}\end{pmatrix}}}} & (12)\end{matrix}$

As described with respect to Eq. 4, the signals {circumflex over(R)}_(X)(t−T) and {circumflex over (R)}_(Y)(t−T) denote preliminarydigital drive signal estimates at the first timeslot, t−T, while thesignals {circumflex over (R)}_(X)(t) and {circumflex over (R)}_(Y)(t)denote preliminary digital drive signal estimates at the secondtimeslot, t.

Given an impairment at the transmitter which affects the I and Qcomponents of the X and Y polarizations differently, it may be shownthat implementation of a dimensional transformation comprising thematrix transformation H₃ may average the impairment across thedimensions.

Example 4

According to a fourth example, a dimensional transformation comprising amodified version of the matrix transformation H₂ of the second examplemay be implemented at a transmitter. In this fourth example, the matrixtransformation, denoted H₄, is provided by Eq. 13:

$\begin{matrix}{H_{4}\overset{\Delta}{=}{e^{\frac{j\; \pi}{4}}\begin{pmatrix}1 & {1j} \\{1j} & 1\end{pmatrix}}} & (13)\end{matrix}$

The matrix transformation H₄ may be used in place of the matrixtransformation H₂ in Eq. 6, thereby resulting in Eq. 14:

$\begin{matrix}{\begin{pmatrix}{S_{X}\left( {t - T} \right)} \\{S_{Y}(t)}\end{pmatrix} = {{H_{4}\begin{pmatrix}{{\hat{S}}_{X}(t)} \\{{\hat{S}}_{Y}(t)}\end{pmatrix}} = {{e^{\frac{j\; \pi}{4}}\begin{pmatrix}1 & {1j} \\{1j} & 1\end{pmatrix}}\begin{pmatrix}{{\hat{S}}_{X}(t)} \\{{\hat{S}}_{Y}(t)}\end{pmatrix}}}} & (14)\end{matrix}$

As described with respect to Eq. 6, the signal S_(X)(t−T) denotes thetransformed digital drive signal in the X polarization at the firsttimeslot, t−T, while the signal S_(Y)(t) denotes the transformed digitaldrive signal in the Y polarization at the second timeslot, t.

According to the fourth example, the inverse dimensional transformationcomprises an inverse matrix transformation H₄ ⁻¹ that is provided in Eq.15:

$\begin{matrix}{H_{4}^{- 1}\overset{\Delta}{=}{e^{\frac{j\; \pi}{4}}\begin{pmatrix}1 & {{- 1}j} \\{{- 1}j} & 1\end{pmatrix}}} & (15)\end{matrix}$

The inverse matrix transformation H₄ ⁻¹ may be used in place of theinverse matrix transformation H₂ ⁻¹ in Eq. 8, thereby resulting in Eq.16:

$\begin{matrix}{\begin{pmatrix}{{\hat{R}}_{X}(t)} \\{{\hat{R}}_{Y}(t)}\end{pmatrix} = {{H_{4}^{- 1}\begin{pmatrix}{R_{X}\left( {t - T} \right)} \\{R_{Y}(t)}\end{pmatrix}} = {{e^{- \frac{j\; \pi}{4}}\begin{pmatrix}1 & {{- 1}j} \\{{- 1}j} & 1\end{pmatrix}}\begin{pmatrix}{R_{X}\left( {t - T} \right)} \\{R_{Y}(t)}\end{pmatrix}}}} & (16)\end{matrix}$

As described with respect to Eq. 8, the signals {circumflex over(R)}_(X)(t) and {circumflex over (R)}_(Y)(t) denote preliminary digitaldrive signal estimates at a first timeslot, t, which are dependent onthe received digital signals R_(X)(t−T) and R_(Y)(t) at the firsttimeslot, t−T, and a second timeslot, t, respectively.

Given an impairment at the transmitter which affects the I and Qcomponents of the X and Y polarizations differently, it may be shownthat implementation of a dimensional transformation comprising thematrix transformation H₄ may average the impairment across thedimensions.

Example 5

According to a fifth example, a dimensional transformation may comprisethe application of a 4×4 Hadamard matrix to the four-dimensional signalŜ_(X)(t), Ŝ_(Y)(t), followed by interleaving different dimensions, forexample, Ŝ_(XI)(t), Ŝ_(XQ)(t), Ŝ_(YI)(t), Ŝ_(YQ)(t). At a transmitter, a4×4 real matrix multiplied with a matrix such as a Hadamard matrix maybe applied, which would differ from the 2×2 complex matrix referred toin Eq. 6. The interleaving of different dimensions may be achieved usingan additional matrix which is also substantially linear, andsubstantially unitary. As a result of these two matrix transformations,the preliminary digital drive signals may be converted into transformeddigital drive signals.

At a receiver, de-interleaving may be applied to received digitalsignals by applying the inverse of the interleaving matrix, followed bythe application of the inverse real Hadamard matrix transformation, suchas the inverse matrix transformation H₂ ⁻¹ provided in Eq. 7. As aresult of these two inverse matrix transformations, together referred toas the inverse dimensional transformation, the received digital signalsmay be converted into preliminary digital drive signal estimates. Thisinverse dimensional transformation may have an advantageous effect onthe distribution of nonlinear noise in the preliminary digital drivesignal estimates.

FIG. 15 is a histogram of received values which have undergone aninverse dimensional transformation incorporating de-interleaving ofdimensions and application of the inverse of the real 4×4 Hadamardmatrix as described with respect to Example 5. The received values arebased on multi-bit symbols having been generated at a transmitter basedon a dual-polarization (DP)-16QAM modulation scheme, the symbols havingundergone a dimensional transformation at the transmitter.

Each of the horizontal and vertical axes shows a specific dimension intime such as {circumflex over (R)}_(XI)(t), {circumflex over(R)}_(XQ)(t), {circumflex over (R)}_(YI)(t), {circumflex over(R)}_(YQ)(t). The received histogram includes a population of clouds ofreceived symbols with centers at the ideal transmitted symbols. Thedifference between the received points and the closest ideal DP-16QAMpoint determines the channel noise. The dotted horizontal and verticallines represent the directions in which the square of the minimumEuclidean distance is equal to one, that is d² _(min)=1. The soliddiagonal arrows represent the direction in which d² _(min)=2. The plotof FIG. 15 demonstrates that applying the inverse dimensionaltransformation distributed the nonlinear noise more along the diagonallines, and less along the vertical and horizontal lines. By causing thenonlinear noise to be distributed in this manner, the likelihood ofdetecting the wrong symbol during carrier recovery may be reduced, whichmay ultimately lead to lower BERs.

In the preceding examples, soft FEC decoding, such as ML decoding, maybe used to recover corrected client data bits. Soft decoding may beperformed over multiple dimensions. By increasing the dimensionalityover which soft decoding is performed, it may be possible to improveperformance by exploiting correlations, and by using higher dimensionalgeometry in the constellation design. However, this improvement may beat the expense of increased circuit complexity.

In the preceding examples, the delay T is described as being an integernumber of timeslots. More generally, however, the delay T that isincluded as part of a dimensional transformation or an inversedimensional transformation may be a non-integer or fractional number.

The scope of the claims should not be limited by the details set forthin the examples, but should be given the broadest interpretationconsistent with the description as a whole.

1-15. (canceled)
 16. A method performed at an optical transmittercomprising circuitry, digital-to-analog converters,electrical-to-optical modulators, and a beam combiner, the methodcomprising: the circuitry generating preliminary digital drive signalsrepresentative of multi-bit symbols; the circuitry generatingtransformed digital drive signals from the preliminary digital drivesignals, wherein the transformed digital drive signals are designed torepresent each multi-bit symbol using a plurality of first dimensions ofan optical carrier, the first dimensions being distributed over two ormore distinct timeslots, and wherein the preliminary digital drivesignals are designed to represent each multi-bit symbol using aplurality of second dimensions of the optical carrier, the seconddimensions differing from the first dimensions; and generating anoptical signal for transmission over an optical communications channelestablished between the optical transmitter and an optical receiver,comprising the digital-to-analog converters converting the transformeddigital drive signals into respective analog signals; theelectrical-to-optical modulators using the analog signals to modulatepolarized components of the optical carrier to produce modulatedpolarized signals; and the beam combiner combining the modulatedpolarized signals to form the optical signal.
 17. The method as claimedin claim 16, wherein the plurality of second dimensions is less than theplurality of first dimensions.
 18. The method as claimed in claim 16,wherein the two or more timeslots are non-consecutive.
 19. The method asclaimed in claim 16, wherein the first dimensions are distributed overtwo polarizations.
 20. The method as claimed in claim 16, wherein thefirst dimensions are distributed over in-phase (I) and quadrature (Q)components of at least one polarization.
 21. The method as claimed inclaim 16, wherein the transformed digital drive signals are generated byapplying a dimensional transformation to the preliminary digital drivesignals.
 22. The method as claimed in claim 21, wherein the dimensionaltransformation comprises a matrix, and wherein the matrix issubstantially linear and substantially unitary.
 23. The method asclaimed in claim 16, wherein the transformed digital drive signals aregenerated from the preliminary digital drive signals using alook-up-table.
 24. The method as claimed in claim 16, furthercomprising: the circuitry applying frequency-domain processing to thetransformed digital drive signals.
 25. The method as claimed in claim24, wherein the frequency-domain processing comprises applying a matchedfilter to the transformed digital drive signals.
 26. An opticaltransmitter configured to generate an optical signal for transmissionover an optical communications channel established between the opticaltransmitter and an optical receiver, the optical transmitter comprising:circuitry configured to generate preliminary digital drive signalsrepresentative of multi-bit symbols; and to apply a dimensionaltransformation to the preliminary digital drive signals to generatetransformed digital drive signals, wherein the transformed digital drivesignals are designed to represent each multi-bit symbol using aplurality of first dimensions of an optical carrier, the firstdimensions being distributed over two or more distinct timeslots, andwherein the preliminary digital drive signals are designed to representeach multi-bit symbol using a plurality of second dimensions of theoptical carrier, the second dimensions differing from the firstdimensions; digital-to-analog converters configured to convert thetransformed digital drive signals into respective analog signals;electrical-to-optical modulators configured to use the analog signals tomodulate polarized components of the optical carrier to producemodulated polarized signals; and a beam combiner configured to combinethe modulated polarized signals to form the optical signal.
 27. Theoptical transmitter as claimed in claim 26, wherein the plurality ofsecond dimensions is less than the plurality of first dimensions. 28.The optical transmitter as claimed in claim 26, wherein the two or moretimeslots are non-consecutive.
 29. The optical transmitter as claimed inclaim 26, wherein the first dimensions are distributed over twopolarizations.
 30. The optical transmitter as claimed in claim 26,wherein the first dimensions are distributed over in-phase (I) andquadrature (Q) components of at least one polarization.
 31. The opticaltransmitter as claimed in claim 26, wherein the transformed digitaldrive signals are generated by applying a dimensional transformation tothe preliminary digital drive signals.
 32. The optical transmitter asclaimed in claim 31, wherein the dimensional transformation comprises amatrix, and wherein the matrix is substantially linear and substantiallyunitary.
 33. The optical transmitter as claimed in claim 26, wherein thetransformed digital drive signals are generated using a look-up-table.34. The optical transmitter as claimed in claim 26, wherein thecircuitry is further configured to apply frequency-domain processing tothe transformed digital drive signals.
 35. The optical transmitter asclaimed in claim 34, wherein the frequency-domain processing comprisesapplying a matched filter to the transformed digital drive signals.